Green's function of the problem of bounded solutions in the case of a block triangular coefficient
| Autor: | Kurbatov, V. G., Kurbatova, I. V. |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums of special convolutions of Green's functions of diagonal blocks of $A$. Comment: 18 pages, 1 figure |
| Databáze: | arXiv |
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